Patterns in surface driven flows

Abstract

Flow in a rectangular basin driven by a surface force is considered. The problem is motivated by flow in geophysical bodies of water driven by wind at the water surface. Results are obtained via numerical computations of the Navier–Stokes equations assuming constant density. The numerical integration is achieved with a splitting method, with Crank–Nicolson for the linear terms, and Adams–Bashforth for the nonlinear terms. Spatial derivatives are treated with finite differences. The forcing has a sinusoidal variation across the top with a sequence of length scales. The results show a symmetric steady stable flow for small Reynolds numbers. As the Reynolds number is increased, the system experiences either a subcritical or supercritical pitchfork bifurcation to an asymmetric steady stable flow, or a local Hopf bifurcation, depending on the aspect ratio of the container and the length scale of the forcing. The asymmetric flow is cellular for forcing length scales commensurate with the depth. For smaller forcing length scales, the asymmetric flow has a basin-filling character at the bottom portion of the basin.